Complexity-Approximation Trade-Offs in Exchange Mechanisms: AMMs vs. LOBs

This paper presents a general framework for the design and analysis of exchange mechanisms between two assets that unifies and enables comparisons between the two dominant paradigms for exchange, constant function market markers (CFMMs) and limit order books (LOBs). In our framework, each liquidity provider (LP) submits to the exchange a downward-sloping demand curve, specifying the quantity of the risky asset it wishes to hold at each price; the exchange buys and sells the risky asset so as to satisfy the aggregate submitted demand. In general, such a mechanism is budget-balanced (i.e., it stays solvent and does not make or lose money) and enables price discovery (i.e., arbitrageurs are incentivized to trade until the exchange's price matches the external market price of the risky asset). Different exchange mechanisms correspond to different restrictions on the set of acceptable demand curves. The primary goal of this paper is to formalize an approximation-complexity trade-off that pervades the design of exchange mechanisms. For example, CFMMs give up expressiveness in favor of simplicity: the aggregate demand curve of the LPs can be described using constant space (the liquidity parameter), but most demand curves cannot be well approximated by any function in the corresponding single-dimensional family. LOBs, intuitively, make the opposite trade-off: any downward-slowing demand curve can be well approximated by a collection of limit orders, but the space needed to describe the state of a LOB can be large. This paper introduces a general measure of exchange complexity, defined by the minimal set of basis functions that generate, through their conical hull, all of the demand functions allowed by an exchange. With this complexity measure in place, we investigate the design of optimally expressive exchange mechanisms, meaning the lowest complexity mechanisms that allow for arbitrary downward-sloping demand curves to be approximated to within a given level of precision. Our results quantify the fundamental trade-off between simplicity and expressivity in exchange mechanisms. As a case study, we interpret the complexity-approximation trade-offs in the widely-used Uniswap v3 AMM through the lens of our framework.